Concordance and mutation

Abstract

We provide a framework for studying the interplay between concordance and positive mutation and identify some of the basic structures relating the two.

The fundamental result in understanding knot concordance is the structure theorem proved by Levine: for $n>1$ there is an isomorphism $\varphi$ from the concordance group ${\mathcal{C}}_n$ of knotted $\left(2n-1\right)$–spheres in $S^{2n+1}$ to an algebraically defined group ${\mathcal{G}}_{\pm }$; furthermore, ${\mathcal{G}}_{\pm }$ is isomorphic to the infinite direct sum ${\mathbb{Z}}^{\infty }\oplus {\mathbb{Z}}_2^{\infty }\oplus {\mathbb{Z}}_4^{\infty }$. It was a startling consequence of the work of Casson and Gordon that in the classical case the kernel of $\varphi$ on ${\mathcal{C}}_1$ is infinitely generated. Beyond this, little has been discovered about the pair $\left({\mathcal{C}}_1,\varphi \right)$.

In this paper we present a new approach to studying ${\mathcal{C}}_1$ by introducing a group, $\mathcal{ℳ}$, defined as the quotient of the set of knots by the equivalence relation generated by concordance and positive mutation, with group operation induced by connected sum. We prove there is a factorization of $\varphi$, ${\mathcal{C}}_1\stackrel{{\varphi }_1}{\to }\mathcal{ℳ}\stackrel{{\varphi }_2}{\to }{\mathcal{G}}_{-}$. Our main result is that both maps have infinitely generated kernels.

Among geometric constructions on classical knots, the most subtle is positive mutation. Positive mutants are indistinguishable using classical abelian knot invariants as well as by such modern invariants as the Jones, Homfly or Kauffman polynomials. Distinguishing positive mutants up to concordance is a far more difficult problem; only one example has been known until now. The results in this paper provide, among other results, the first infinite families of knots that are distinct from their positive mutants, even up to concordance.